monte carlo in different ensembles chapter 5 nvt ensemble npt ensemble grand-canonical ensemble...

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epartm entofChem ical Engine Monte Carlo in different ensembles Chapter 5 NVT ensemble NPT ensemble Grand-canonical ensemble Exotic ensembles

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Department of Chemical Engineering

Monte Carlo in different ensemblesChapter 5

NVT ensembleNPT ensemble

Grand-canonical ensembleExotic ensembles

Department of Chemical Engineering2

Statistical Thermodynamics

NVTQF ln

NNNN

NVTNVT

UANQ

A rexprdr!

113

NNNNVT UN

Q rexpdr!

13

Partition function

Ensemble average

Free energy

3

1 1N r dr' r' r exp r' exp r

!N N N N N N

NNVT

U UQ N

Probability to find a particular configuration

Department of Chemical Engineering4

Detailed balance

acc( ) ( ) ( ) ( )

acc( ) ( ) ( ) ( )

o n N n n o N n

n o N o o n N o

( ) ( )K o n K n o ( ) ( ) ( ) acc( )K o n N o o n o n

o n

( ) ( ) ( ) acc( )K n o N n n o n o

Department of Chemical Engineering5

NVT-ensemble

acc( ) ( )

acc( ) ( )

o n N n

n o N o

( ) expN n U n

acc( )exp

acc( )

o nU n U o

n o

Department of Chemical Engineering6

Department of Chemical Engineering7

NPT ensemble

We control the temperature, pressure, and number of particles.

Department of Chemical Engineering8

Scaled coordinates

/i i Ls r

NNNNVT UN

Q rexpdr!

13

Partition function

Scaled coordinates

This gives for the partition function

3

3

3

ds exp s ;!

ds exp s ;!

NN N

NVT N

NN N

N

LQ U L

N

VU L

N

The energy depends on the real coordinates

Department of Chemical Engineering9

The perfect simulation ensemble

Here they are an ideal gas

Here they interact

What is the statistical thermodynamics of this ensemble?

Department of Chemical Engineering10

The perfect simulation ensemble: partition function

3ds exp s ;

!

NN N

NVT N

VQ U L

N

0

0, , 03 3

ds exp s ;! !

ds exp s ;

M N NM N M N

MV NV T M N N

N N

V V VQ U L

M N N

U L

0

0, , 3 3

ds exp s ;! !

M N NN N

MV NV T M N N

V V VQ U L

M N N

Department of Chemical Engineering11

0

0, , 3 3

ds exp s ;! !

M N NN N

MV NV T M N N

V V VQ U L

M N N

To get the Partition Function of this system, we have to integrate over all possible volumes:

0

0, , 3 3

d ds exp s ;! !

M N NN N

MV N T M N N

V V VQ V U L

M N N

Now let us take the following limits:

0

constantM MV V

As the particles are an ideal gas in the big reservoir we have:

P

Department of Chemical Engineering12

We have

0

0, , 3 3

d ds exp s ;! !

M N NN N

MV N T M N N

V V VQ V U L

M N N

0 0 0 0 01 expM N M NM N M NV V V V V V M N V V

0 0 0exp expM N M N M NV V V V V PV

This gives:

3d exp ds exp s ;

!N N N

NPT N

PQ V PV V U L

N

To make the partition function dimension less

Department of Chemical Engineering13

NPT EnsemblePartition function:

3d exp ds exp s ;

!N N N

NPT N

PQ V PV V U L

N

Probability to find a particular configuration:

, exp exp s ;N N NNPTN V V PV U L s

Sample a particular configuration:• Change of volume • Change of reduced coordinates

Acceptance rules ??

Detailed balance

Department of Chemical Engineering14

Detailed balance

acc( ) ( ) ( ) ( )

acc( ) ( ) ( ) ( )

o n N n n o N n

n o N o o n N o

( ) ( )K o n K n o ( ) ( ) ( ) acc( )K o n N o o n o n

o n

( ) ( ) ( ) acc( )K n o N n n o n o

Department of Chemical Engineering15

NPT-ensemble

acc( ) ( )

acc( ) ( )

o n N n

n o N o

, exp exp s ;N N NNPTN V V PV U L s

Suppose we change the position of a randomly selected particle

Nn

No

exp exp s ;acc( )

acc( ) exp exp s ;

N

N

V PV U Lo n

n o V PV U L

Nn

No

exp s ;exp

exp s ;

U LU n U o

U L

Department of Chemical Engineering16

NPT-ensemble

acc( ) ( )

acc( ) ( )

o n N n

n o N o

, exp exp s ;N N NNPTN V V PV U L s

Suppose we change the volume of the system

N

N

exp exp s ;acc( )

acc( ) exp exp s ;

Nn n n

No o o

V PV U Lo n

n o V PV U L

exp exp 0

N

nn o

o

VP V V U n U

V

Department of Chemical Engineering17

Algorithm: NPT

• Randomly change the position of a particle

• Randomly change the volume

Department of Chemical Engineering18

Department of Chemical Engineering19

Department of Chemical Engineering20

Department of Chemical Engineering21

NPT simulations

Department of Chemical Engineering22

Grand-canonical ensemble

What are the equilibrium conditions?

Department of Chemical Engineering23

Grand-canonical ensemble

We impose:– Temperature– Chemical potential– Volume

– But NOT pressure

Department of Chemical Engineering24

The Murfect ensemble

Here they are an ideal gas

Here they interact

What is the statistical thermodynamics of this ensemble?

Department of Chemical Engineering25

The Murfect simulation ensemble: partition function

3ds exp s ;

!

NN N

NVT N

VQ U L

N

0

0, , 03 3

ds exp s ;! !

ds exp s ;

M N NM V M N

MV NV T M N N

N N

V V VQ U L

M N N

U L

0

0, , 3 3

ds exp s ;! !

M N NN N

MV NV T M N N

V V VQ U L

M N N

Department of Chemical Engineering26

0

0, , 3 3

ds exp s ;! !

M N NN N

MV NV T M N N

V V VQ U L

M N N

To get the Partition Function of this system, we have to sum over all possible number of particles

0

0, , 3 3

0

ds exp s ;! !

M N NN MN N

MV N T M N NN

V V VQ U L

M N N

Now let us take the following limits:

0

constantM MV V

As the particles are an ideal gas in the big reservoir we have:

3lnBk T

30

expds exp s ;

!

NNN N

VT NN

N VQ U L

N

Department of Chemical Engineering27

MuVT EnsemblePartition function:

Probability to find a particular configuration:

3

exp, exp s ;

!

NN N

VT N

N VN V U L

N

s

Sample a particular configuration:• Change of the number of particles• Change of reduced coordinates

Acceptance rules ??

Detailed balance

30

expds exp s ;

!

NNN N

VT NN

N VQ U L

N

Department of Chemical Engineering28

Detailed balance

acc( ) ( ) ( ) ( )

acc( ) ( ) ( ) ( )

o n N n n o N n

n o N o o n N o

( ) ( )K o n K n o ( ) ( ) ( ) acc( )K o n N o o n o n

o n

( ) ( ) ( ) acc( )K n o N n n o n o

Department of Chemical Engineering29

VT-ensemble

acc( ) ( )

acc( ) ( )

o n N n

n o N o

Suppose we change the position of a randomly selected particle

Nn3

No3

expexp s ;acc( ) !

expacc( )exp s ;

!

N

N

N

N

N VU Lo n N

N Vn oU L

N

exp 0U n U

3

exp, exp s ;

!

NN N

VT N

N VN V U L

N

s

Department of Chemical Engineering30

VT-ensemble

acc( ) ( )

acc( ) ( )

o n N n

n o N o

Suppose we change the number of particles of the system

1N+1

3 3

N3

exp 1exp s ;

1 !acc( )

expacc( )exp s ;

!

N

nN

N

oN

N VU L

No n

N Vn oU L

N

3

exp, exp s ;

!

NN N

VT N

N VN V U L

N

s

3

expexp

1

VU

N

Department of Chemical Engineering31

Department of Chemical Engineering32

Department of Chemical Engineering33

Application: equation of state of Lennard-Jones

Department of Chemical Engineering34

Application: adsorption in zeolites

Department of Chemical Engineering35

Exotic ensembles

What to do with a biological membrane?

Department of Chemical Engineering36

Model membrane: Lipid bilayer

hydrophilic head group

two hydrophobic tails

water

water

Department of Chemical Engineering37

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Questions

• What is the surface tension of this system?

• What is the surface tension of a biological membrane?

• What to do about this?

Department of Chemical Engineering39

Phase diagram: alcohol

Department of Chemical Engineering40

Simulations at imposed surface tension

• Simulation to a constant surface tension– Simulation box: allow the area of the bilayer to

change in such a way that the volume is constant.

Department of Chemical Engineering41

Constant surface tension simulation

A)]})(A'A);(U)A';(U[exp{min(1, NNaccP ss

AUF

AA’

LL’

A L = A’ L’ = V

, exp[ (U( ) A)]N NN T A r rN

Department of Chemical Engineering42

(Ao) = -0.3 +/- 0.6

(Ao) = 2.5 +/- 0.3

(Ao) = 2.9 +/- 0.3

Tensionless state: = 0